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Register a new userA low-communication-overhead parallel method for the 3D incompressible Navier-Stokes equations
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This paper presents a low-communication-overhead parallel method for solving the 3D incompressible Navier-Stokes equations. A fully-explicit projection method with second-order space-time accuracy is adopted. Combined with fast Fourier transforms, the parallel diagonal dominant (PDD) algorithm for the tridiagonal system is employed to solve the pressure Poisson equation, differing from its recent applications to compact scheme derivatives computation (Abide et al. 2017) and alternating-direction-implicit method (Moon et al. 2020). The number of all-to-all communications is decreased to only two, in a 2D pencil-like domain decomposition. The resulting MPI/OpenMP hybrid parallel code shows excellent strong scalability up to $10^4$ cores and small wall-clock time per timestep. Numerical simulations of turbulent channel flow at different friction Reynolds numbers ($Re_{tau}$ = 550, 1000, 2000) have been conducted and the statistics are in good agreement with the reference data. The proposed method allows massively simulation of wall turbulence at high Reynolds numbers as well as many other incompressible flows.
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There have been several efforts to Physics-informed neural networks (PINNs) in the solution of the incompressible Navier-Stokes fluid. The loss function in PINNs is a weighted sum of multiple terms, including the mismatch in the observed velocity and pressure data, the boundary and initial constraints, as well as the residuals of the Navier-Stokes equations. In this paper, we observe that the weighted combination of competitive multiple loss functions plays a significant role in training PINNs effectively. We establish Gaussian probabilistic models to define the loss terms, where the noise collection describes the weight parameter for each loss term. We propose a self-adaptive loss function method, which automatically assigns the weights of losses by updating the noise parameters in each epoch based on the maximum likelihood estimation. Subsequently, we employ the self-adaptive loss balanced Physics-informed neural networks (lbPINNs) to solve the incompressible Navier-Stokes equations,hspace{-1pt} includinghspace{-1pt} two-dimensionalhspace{-1pt} steady Kovasznay flow, two-dimensional unsteady cylinder wake, and three-dimensional unsteady Beltrami flow. Our results suggest that the accuracy of PINNs for effectively simulating complex incompressible flows is improved by adaptively appropriate weights in the loss terms. The outstanding adaptability of lbPINNs is not irrelevant to the initialization choice of noise parameters, which illustrates the robustness. The proposed method can also be employed in other problems where PINNs apply besides fluid problems.
Whether the 3D incompressible Navier-Stokes equations can develop a finite time singularity from smooth initial data is one of the most challenging problems in nonlinear PDEs. In this paper, we present some new numerical evidence that the 3D incompressible axisymmetric Navier-Stokes equations with smooth initial data of finite energy develop nearly singular solutions at the origin. This nearly singular behavior is induced by a potential finite time singularity of the 3D Euler equations that we reported in cite{Hou-euler-2021}. One important feature of the potential Euler singularity is that the solution develops nearly self-similar scaling properties that are compatible with those of the 3D Navier-Stokes equations. We will present numerical evidence that the 3D Navier-Stokes equations develop nearly singular scaling properties with maximum vorticity increased by a factor of $10^7$. Moreover, the nearly self-similar profiles seem to be very stable to the small perturbation of the initial data. However, the 3D Navier-Stokes equations with our initial data do not develop a finite time singularity due to the development of a mild two-scale structure in the late stage, which eventually leads to viscous dominance over vortex stretching. To maintain the balance between the vortex stretching term and the diffusion term, we solve the 3D Navier-Stokes equations with a time-dependent viscosity roughly of order $O(|log(T-t)|^{-3})$ in the late stage. We present strong numerical evidence that the 3D Navier-Stokes equations with such time-dependent viscosity develop a finite time singularity.
A conservative discretization of incompressible Navier-Stokes equations is developed based on discrete exterior calculus (DEC). A distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product. The governing equations are first rewritten using the exterior calculus notation, replacing vector calculus differential operators by the exterior derivative, Hodge star and wedge product operators. The discretization is then carried out by substituting with the corresponding discrete operators based on the DEC framework. Numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured-triangular meshes, and first order accuracy for otherwise unstructured meshes. By construction, the method is conservative in that both mass and vorticity are conserved up to machine precision. The relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step.
We investigate the spatio-temporal structure of the most likely configurations realising extremely high vorticity or strain in the stochastically forced 3D incompressible Navier-Stokes equations. Most likely configurations are computed by numerically finding the highest probability velocity field realising an extreme constraint as solution of a large optimisation problem. High-vorticity configurations are identified as pinched vortex filaments with swirl, while high-strain configurations correspond to counter-rotating vortex rings. We additionally observe that the most likely configurations for vorticity and strain spontaneously break their rotational symmetry for extremely high observable values. Instanton calculus and large deviation theory allow us to show that these maximum likelihood realisations determine the tail probabilities of the observed quantities. In particular, we are able to demonstrate that artificially enforcing rotational symmetry for large strain configurations leads to a severe underestimate of their probability, as it is dominated in likelihood by an exponentially more likely symmetry broken vortex-sheet configuration.
A dynamic procedure for the Lagrangian Averaged Navier-Stokes-$alpha$ (LANS-$alpha$) equations is developed where the variation in the parameter $alpha$ in the direction of anisotropy is determined in a self-consistent way from data contained in the simulation itself. The dynamic model is initially tested in forced and decaying isotropic turbulent flows where $alpha$ is constant in space but it is allowed to vary in time. It is observed that by using the dynamic LANS-$alpha$ procedure a more accurate simulation of the isotropic homogeneous turbulence is achieved. The energy spectra and the total kinetic energy decay are captured more accurately as compared with the LANS-$alpha$ simulations using a fixed $alpha$. In order to evaluate the applicability of the dynamic LANS-$alpha$ model in anisotropic turbulence, a priori test of a turbulent channel flow is performed. It is found that the parameter $alpha$ changes in the wall normal direction. Near a solid wall, the length scale $alpha$ is seen to depend on the distance from the wall with a vanishing value at the wall. On the other hand, away from the wall, where the turbulence is more isotropic, $alpha$ approaches an almost constant value. Furthermore, the behavior of the subgrid scale stresses in the near wall region is captured accurately by the dynamic LANS-$alpha$ model. The dynamic LANS-$alpha$ model has the potential to extend the applicability of the LANS-$alpha$ equations to more complicated anisotropic flows.
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